# `independent_array_joint`
$$
p(\mathbf{y} \mid \mathbf{x}) = \prod_{i=1}^n p(y_i \mid x_i)
$$
Log
$$
\log{p(\mathbf{y} \mid \mathbf{x})} = \log {p(y_1 \mid x_1)} + \cdots + \log{p(y_i \mid x_1)}+ \cdots + \log {p(y_n \mid x_1)}
$$
Log Value Difference
$$
\frac{\partial \log{p(\mathbf{y} \mid \mathbf{x})}}{\partial \mathbf{y}}
= \begin{bmatrix} \frac{\partial \log {p(y_1 \mid x_1)}}{\partial y_1} + \cdots + \frac{\partial \log {p(y_n \mid x_n)}}{\partial y_1} \\ \vdots \\ \frac{\partial \log {p(y_1 \mid x_1)}}{\partial y_i} + \cdots + \frac{\partial \log {p(y_n \mid x_n)}}{\partial y_i} \\ \vdots \\ \frac{\partial \log {p(y_1 \mid x_1)}}{\partial y_n} + \cdots + \frac{\partial \log {p(y_n \mid x_n)}}{\partial y_n} \end{bmatrix}
\\
= \begin{bmatrix} \frac{\partial \log {p(y_1 \mid x_1)}}{\partial y_1} + 0 + \cdots \\ \vdots \\ \cdots + 0 + \frac{\partial \log {p(y_i \mid x_i)}}{\partial y_i} + 0 + \cdots \\ \vdots \\ 0 + \cdots + \frac{\partial \log {p(y_n \mid x_n)}}{\partial y_n} \end{bmatrix}
\\
= \begin{bmatrix} \frac{\partial \log {p(y_1 \mid x_1)}}{\partial y_1} \\ \vdots \\ \frac{\partial \log {p(y_i \mid x_i)}}{\partial y_i} \\ \vdots \\ \frac{\partial \log {p(\mathbf{y}_n \mid x_n)}}{\partial y_n} \end{bmatrix}
$$
Log Condition Difference
$$
\frac{\partial \log{p(\mathbf{y} \mid \mathbf{x})}}{\partial \mathbf{x}}
= \begin{bmatrix} \frac{\partial \log {p(y_1 \mid x_1)}}{\partial x_1} + \cdots + \frac{\partial \log {p(y_n \mid x_n)}}{\partial x_1} \\ \vdots \\ \frac{\partial \log {p(y_1 \mid x_1)}}{\partial x_i} + \cdots + \frac{\partial \log {p(y_n \mid x_n)}}{\partial x_i} \\ \vdots \\ \frac{\partial \log {p(y_1 \mid x_1)}}{\partial x_n} + \cdots + \frac{\partial \log {p(y_n \mid x_n)}}{\partial x_n} \end{bmatrix}
\\
= \begin{bmatrix} \frac{\partial \log {p(y_1 \mid x_1)}}{\partial x_1} + 0 + \cdots \\ \vdots \\ \cdots + 0 + \frac{\partial \log {p(y_i \mid x_i)}}{\partial x_i} + 0 + \cdots \\ \vdots \\ 0 + \cdots + \frac{\partial \log {p(y_n \mid x_n)}}{\partial x_n} \end{bmatrix}
\\
=\begin{bmatrix} \frac{\partial \log {p(y_1 \mid x_1)}}{\partial x_1} \\ \vdots \\ \frac{\partial \log {p(y_i \mid x_i)}}{\partial x_i} \\ \vdots \\ \frac{\partial \log {p(\mathbf{y}_n \mid x_n)}}{\partial x_n} \end{bmatrix}
$$